# Algebra Seminar 2016

Current contacts: Vasily Dolgushev, Ed Letzter, Martin Lorenz or Chelsea Walton

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Monday January 25, 2016 at 13:30, Wachman 617

Martin Lorenz, Temple University.

• Monday February 1, 2016 at 13:30, Wachman 617

Martin Lorenz, Temple University.

• Monday February 8, 2016 at 13:30, Wachman 617

Martin Lorenz, Temple University.

• Monday February 15, 2016 at 13:30, Wachman 617

• Monday February 22, 2016 at 13:30, Wachman 617

Maria Sabitova, CUNY.

• Monday February 29, 2016 at 13:30, Wachman 617
no seminar: spring break

• Monday March 7, 2016 at 13:30, Wachman 617

Chelsea Walton, Temple University.

• Monday March 14, 2016 at 13:30, Wachman 617

Chelsea Walton, Temple University.

• Monday March 21, 2016 at 13:30, Wachman 617
--- talk cancelled; no seminar today ---

• Friday March 25, 2016 at 15:00, Wachman 617

-Note different day and time- Sonia Natale, University of Cordoba.

• Monday March 28, 2016 at 13:30, Wachman 617

Xingting Wang, Temple University

• Monday April 4, 2016 at 13:30, Wachman 617

Xingting Wang, Temple University.

• Monday April 11, 2016 at 13:30, Wachman 617

Marton Hablicsek, University of Pennsylvania.

• Monday September 12, 2016 at 13:30, Wachman 617
Graphs on Surfaces. Introduction

Vasily Dolgushev, Temple University

This is the first lecture in the mini-course on dessins d'enfant (child's drawings) and the Grothendieck-Teichmueller group GT. In this lecture, I will introduce the "main characters" of the story and say a few words about the motivation. In the remaining lectures of this mini-course we will talk about the material of the first two chapters of the book "Graphs on surfaces and their applications" by Lando, Zagier and Zvonkin.

• Monday September 19, 2016 at 13:30, Wachman Hall Rm. 617
Everything about constellations... Well, almost everything

Vasily Dolgushev, Temple University

A constellation is a sequence of permutations in S_n satisfying some conditions. We will talk about the cartographic group of a constellation, isomorphic (and conjugate) constellations. Finally, we will show that the braid group B_k acts on constellations of length k.

• Monday September 26, 2016 at 13:30, Wachman Hall, Rm. 617
A "crash course" on covering spaces

Vasily Dolgushev, Temple University

I will review the notion of the fundamental group and the notion of the covering space. I will briefly outline the classification of connected covering spaces over a connected base. The notion of the monodromy group and the notion of a normal covering will be discussed. Some examples will be given. This talk is a part of the mini-course "Graphs on Surfaces".

• Monday October 3, 2016 at 13:30, Wachman Hall, Rm. 617
A "crash course" on covering spaces, II

Vasily Dolgushev, Temple University

I plan to finish the brief review of covering spaces. This talk is a part of the mini-course "Graphs on Surfaces".

• Monday October 10, 2016 at 13:30, Wachman Hall Rm. 617
Constellations versus coverings of the punctured sphere

Vasily Dolgushev, Temple University

I will talk about the correspondence between constellations of length k and connected coverings of the sphere with k punctures. I will recall the Riemann-Hurwitz formula and use it to compute the genus of the covering surface corresponding to the constellation. This talk is a part of the mini-course "Graphs on Surfaces".

• Monday October 17, 2016 at 13:30, Wachman Hall, Rm. 617
Connections between Demazure flags, Chebyshev polynomials and mock theta functions

Rekha Biswal, The Institute of Mathematical Sciences, Chennai, India

In this talk, we will explore some (surprising) connections between representation theory, combinatorics and number theory. We are interested in studying a family of finite dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $A_2^{(2)}$. These families of modules are of a lot of interest because of their connections to the representation theory of quantum affine algebras. In a joint work with Vyjayanthi Chari and Deniz Kus, it is proved that these modules admit a decreasing filtration whose successive quotients are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level l-Demazure module admits a filtration by level m Demazure modules for $m > l-1$. In this talk, we shall discuss the generating functions which encode the multiplicity of a given Demazure module and prove that the generating functions of graded multiplicities define hypergeometric series and that they are related to $q$-Fibonacci polynomials defined by Carlitz in the case when $l = 1, 2$ and $m = 2, 3$. We will also see that the generating functions of numerical multiplicities are related to Chebyshev polynomials of second kind and the generating functions of graded multiplicities of Demazure modules in local Weyl modules relates to Ramanujan’s fifth order mock theta functions in certain special cases.

• Monday October 24, 2016 at 13:30, Wachman Hall Rm. 617
The relation between unimodularity and Calabi-Yau property for Poisson algebras

Xingting Wang, Temple University

Poisson geometry is originated in classical mechanics where one describes the time evolution of a mechanical system by solving Hamilton's equations in terms of the Hamiltonian vector field. Recently, the development of Poisson geometry has deeply entangled with noncommutative algebra and noncommutative geometry.

In this talk, I will introduce Poisson (co)homology using Poisson enveloping algebras. I will explain the unimodularity of Poisson algebras has a close relationship with the Calabi-Yau property of their enveloping algebras. This echoes Dolgushev’s result such that the deformation quantization of a Poisson algebra is a Calabi-Yau algebra if and only if the corresponding Poisson structure is unimodular.

• Monday October 31, 2016 at 13:30, Wachman Hall, Rm. 617
The relation between Calabi-Yau property and unimodularity for Poisson algebras

Xingting Wang, Temple University

This is the continuation of the last week's talk.

• Monday November 7, 2016 at 13:30, Wachman Hall Rm. 617
Hypermaps, constellations, and triangulations of surfaces

Vasily Dolgushev, Temple University

A hypermap is a bipartite graph $G$ "drawn" on an oriented Riemann surface $S$ so that the complement $S \setminus G$ is a disjoint union of (contractible) cells. I will talk about the correspondence between hypermap and constellations of length 3. This talk is a part of the mini-course "Graphs on Surfaces".

• Monday November 14, 2016 at 13:30, Wachman Hall Rm. 617
Hypermaps, constellations and triangulations of surfaces, II

Vasily Dolgushev, Temple University

This is the second talk about hypermaps, constellations and triangulations of surfaces. This talk is a part of the mini-course "Graphs on Surfaces".

• Monday November 28, 2016 at 13:30, Wachman Hall Rm. 617
The derived Picard group of an affine Azumaya algebra

Cris Negron, M.I.T.

I will discuss the derived Picard group of an Azumaya algebra A over an affine scheme X. The derived Picard group is a derived invariant which can be seen as a refined version of the group of auto-equivalences on the derived category of quasi-coherent sheaves over A. I will explain how this group decomposes in terms of the Picard group of X, global sections of the constant sheaf of integers on X, the stabilizer of the Brauer class of A in Aut(X), and a mysterious 2-cocycle taking values in the Picard group. We will follow the basic example of the Weyl algebra in finite characteristic throughout.

• Monday December 12, 2016 at 13:30, Wachman Hall Rm. 617
Belyi pairs versus hypermaps

Vasily Dolgushev, Temple University

A Belyi pair is a pair $(X,f)$ where $X$ is a Riemann surface and $f$ is a holomorphic map from $X$ to the complex projective plane with the critical values $0$, $1$, and $\infty$. I will talk about the correspondence between Belyi pairs and hypermaps. I will also describe several examples. This talk is a part of the mini-course "Graphs on Surfaces".